Advertisement

The propagation delay in the timing of a pulsar orbiting a supermassive black hole

  • Eva HackmannEmail author
  • Arnab Dhani
Editor’s Choice (Research Article)

Abstract

The observation of a pulsar closely orbiting the galactic center supermassive black hole would open the window for an accurate determination of the black hole parameters and for new tests of General Relativity. An important relativistic effect which has to be taken into account in the timing model is the propagation delay of the pulses in the gravitational field of the black hole. Due to the extreme mass ratio of the pulsar and the supermassive back hole we use the test particle limit to derive an exact analytical formula for the propagation delay in a Schwarzschild spacetime. We then compare this result to the propagation delays derived in the usually employed post-Newtonian approximation, in particular to the Shapiro delay up to the second post-Newtonian order. For edge-on orbits we also consider modifications of the Shapiro delay which take the lensing effects into account. Our results are then used to assess the accuracy of the different orders of the post-Newtonian approximation of the propagation delay. This comparison indicates that for (nearly) edge-on orbits the new exact delay formula should be used.

Keywords

Black holes Light propagation Pulsar timing 

Notes

Acknowledgements

The authors thank the research training group GRK 1620 ”Models of Gravity”, funded by the German Research Foundation (DFG), for support. E.H. gratefully acknowledges support from the DFG funded collaborative research center SFB 1128 ”Relativistic geodesy with quantum sensors (geo-Q)”. A.D. is thankful to University of Bremen for its hospitality and support where this work was conceptualized. We thank D. Schwarz and J. Verbiest for fruitful discussions.

References

  1. 1.
    Ghez, A.M., Salim, S., Weinberg, N.N., Lu, J.R., Do, T., Dunn, J.K., Matthews, K., Morris, M.R., Yelda, S., Becklin, E.E., Kremenek, T., Milosavljevic, M., Naiman, J.: Measuring distance and properties of the milky way’s central supermassive black hole with stellar orbits. Astrophys. J. 689, 1044–1062 (2008)ADSCrossRefGoogle Scholar
  2. 2.
    Gillessen, S., Eisenhauer, F., Trippe, S., Alexander, T., Genzel, R., Martins, F., Ott, T.: Monitoring stellar orbits around the massive black hole in the galactic center. Astrophys. J. 692, 1075–1109 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    Lu, R.-S., Broderick, A.E., Baron, F., Monnier, J.D., Fish, V.L., Doeleman, S.S., Pankratius, V.: Imaging the supermassive black hole shadow and jet base of M87 with the event horizon telescope. Astrophys. J. 788, 120 (2014)ADSCrossRefGoogle Scholar
  4. 4.
    Falcke, H., Melia, F., Agol, E.: Viewing the shadow of the black hole at the galactic center. Astrophys. J. Lett. 528, L13–L16 (2000)ADSCrossRefGoogle Scholar
  5. 5.
    Liu, K., Wex, N., Kramer, M., Cordes, J.M., Lazio, T.J.W.: Prospects for probing the spacetime of Sgr A* with pulsars. Astrophys. J. 747, 1 (2012)ADSCrossRefGoogle Scholar
  6. 6.
    Psaltis, D., Wex, N., Kramer, M.: A quantitative test of the no-hair theorem with Sgr A* using stars, pulsars, and the event horizon telescope. Astrophys. J. 818, 121 (2016)ADSCrossRefGoogle Scholar
  7. 7.
    Broderick, A.E., Johannsen, T., Loeb, A., Psaltis, D.: Testing the no-hair theorem with event horizon telescope observations of sagittarius A*. Astrophys. J. 784, 7 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    Goddi, C., Falcke, H., Kramer, M., Rezzolla, L., Brinkerink, C., Bronzwaer, T., Davelaar, J.R.J., Deane, R., de Laurentis, M., Desvignes, G., Eatough, R.P., Eisenhauer, F., Fraga-Encinas, R., Fromm, C.M., Gillessen, S., Grenzebach, A., Issaoun, S., Janßen, M., Konoplya, R., Krichbaum, T.P., Laing, R., Liu, K., Lu, R.-S., Mizuno, Y., Moscibrodzka, M., Müller, C., Olivares, H., Pfuhl, O., Porth, O., Roelofs, F., Ros, E., Schuster, K., Tilanus, R., Torne, P., van Bemmel, I., van Langevelde, H.J., Wex, N., Younsi, Z., Zhidenko, A.: BlackHoleCam: fundamental physics of the galactic center. Int. J. Mod. Phys. D 26, 1730001–239 (2017)ADSCrossRefGoogle Scholar
  9. 9.
    Hees, A., Do, T., Ghez, A.M., Martinez, G.D., Naoz, S., Becklin, E.E., Boehle, A., Chappell, S., Chu, D., Dehghanfar, A., Kosmo, K., Lu, J.R., Matthews, K., Morris, M.R., Sakai, S., Schödel, R., Witzel, G.: Testing general relativity with stellar orbits around the supermassive black hole in our galactic center. Phys. Rev. Lett. 118(21), 211101 (2017)ADSCrossRefGoogle Scholar
  10. 10.
    Grould, M., Meliani, Z., Vincent, F.H., Grandclément, P., Gourgoulhon, E.: Comparing timelike geodesics around a Kerr black hole and a boson star. Class. Quantum Gravity 34(21), 215007 (2017)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Cunha, P.V.P., Herdeiro, C.A.R., Radu, E., Rúnarsson, H.F.: Shadows of Kerr black holes with scalar hair. Phys. Rev. Lett. 115(21), 211102 (2015)ADSCrossRefGoogle Scholar
  12. 12.
    Cunha, P.V.P., Font, J.A., Herdeiro, C., Radu, E., Sanchis-Gual, N., Zilhão, M.: Lensing and dynamics of ultracompact bosonic stars. Phys. Rev. D 96(10), 104040 (2017)ADSCrossRefGoogle Scholar
  13. 13.
    Vincent, F.H., Gourgoulhon, E., Herdeiro, C., Radu, E.: Astrophysical imaging of Kerr black holes with scalar hair. Phys. Rev. D 94(8), 084045 (2016)ADSCrossRefGoogle Scholar
  14. 14.
    Vincent, F.H., Meliani, Z., Grandclément, P., Gourgoulhon, E., Straub, O.: Imaging a boson star at the Galactic center. Class. Quantum Gravity 33(10), 105015 (2016)ADSCrossRefGoogle Scholar
  15. 15.
    Mizuno, Y., Younsi, Z., Fromm, C.M., Porth, O., De Laurentis, M., Olivares, H., Falcke, H., Kramer, M., Rezzolla, L.: The current ability to test theories of gravity with black hole shadows. Nat. Astron. 2, 585 (2018)ADSCrossRefGoogle Scholar
  16. 16.
    Pfahl, E., Loeb, A.: Probing the spacetime around sagittarius A* with radio pulsars. Astrophys. J. 615, 253–258 (2004)ADSCrossRefGoogle Scholar
  17. 17.
    Macquart, J.-P., Kanekar, N., Frail, D.A., Ransom, S.M.: A high-frequency search for pulsars within the central parsec of Sgr A*. Astrophys. J. 715, 939–946 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    Macquart, J.-P., Kanekar, N.: On detecting millisecond pulsars at the galactic center. Astrophys. J. 805, 172 (2015)ADSCrossRefGoogle Scholar
  19. 19.
    Rajwade, K.M., Lorimer, D.R., Anderson, L.D.: Detecting pulsars in the galactic centre. MNRAS 471, 730–739 (2017)ADSCrossRefGoogle Scholar
  20. 20.
    Keane, E., Bhattacharyya, B., Kramer, M., Stappers, B., Keane, E. F., Bhattacharyya, B., Kramer, M., Stappers, B. W., Bates, S. D., Burgay, M., Chatterjee, S., Champion, D. J., Eatough, R. P., Hessels, J. W. T., Janssen, G., Lee, K. J., van Leeuwen, J., Margueron, J., Oertel, M., Possenti, A., Ransom, S., Theureau, G., Torne, P.: Proceedings of Science vol. 215, Advancing Astrophysics with the Square Kilometre Array (AASKA14), id. 40 (2015)Google Scholar
  21. 21.
    Bower, G. C., Chatterjee, S., Cordes, J., Demorest, P., Deneva, J. S., Dexter, J., Kramer, M., Lazio, J., Ransom, S., Shao, L., Wex, N., Wharton, R.: Galactic center pulsars with the ngVLA. In: Murphy,E. (ed.) Science with a Next Generation Very Large Array, Astronomical Society of the Pacific Conference Series, vol. 517, p. 793 (2018)Google Scholar
  22. 22.
    Eisenhauer, F., Perrin, G., Brandner, W., Straubmeier, C., Perraut, K., Amorim, A., Schöller, M., Gillessen, S., Kervella, P., Benisty, M., Araujo-Hauck, C., Jocou, L., Lima, J., Jakob, G., Haug, M., Clénet, Y., Henning, T., Eckart, A., Berger, J.-P., Garcia, P., Abuter, R., Kellner, S., Paumard, T., Hippler, S., Fischer, S., Moulin, T., Villate, J., Avila, G., Gräter, A., Lacour, S., Huber, A., Wiest, M., Nolot, A., Carvas, P., Dorn, R., Pfuhl, O., Gendron, E., Kendrew, S., Yazici, S., Anton, S., Jung, Y., Thiel, M., Choquet, É., Klein, R., Teixeira, P., Gitton, P., Moch, D., Vincent, F., Kudryavtseva, N., Ströbele, S., Sturm, S., Fédou, P., Lenzen, R., Jolley, P., Kister, C., Lapeyrère, V., Naranjo, V., Lucuix, C., Hofmann, R., Chapron, F., Neumann, U., Mehrgan, L., Hans, O., Rousset, G., Ramos, J., Suarez, M., Lederer, R., Reess, J.-M., Rohloff, R.-R., Haguenauer, P., Bartko, H., Sevin, A., Wagner, K., Lizon, J.-L., Rabien, S., Collin, C., Finger, G., Davies, R., Rouan, D., Wittkowski, M., Dodds-Eden, K., Ziegler, D., Cassaing, F., Bonnet, H., Casali, M., Genzel, R., Lena, P.: GRAVITY: observing the universe in motion. Messenger 143, 16–24 (2011)ADSGoogle Scholar
  23. 23.
    Collaboration, Gravity: First light for GRAVITY: a new era for optical interferometry. Messenger 170, 10–15 (2017)ADSGoogle Scholar
  24. 24.
    Verbiest, J.P.W., Bailes, M., van Straten, W., Hobbs, G.B., Edwards, R.T., Manchester, R.N., Bhat, N.D.R., Sarkissian, J.M., Jacoby, B.A., Kulkarni, S.R.: Precision timing of PSR J0437–4715: an accurate pulsar distance, a high pulsar mass, and a limit on the variation of Newton’s gravitational constant. Astrophys. J. 679, 675–680 (2008)ADSCrossRefGoogle Scholar
  25. 25.
    Kramer, M.: Probing gravitation with pulsars. In: Neutron Stars and Pulsars: Challenges and Opportunities after 80 Years, Proceedings of the International Astronomical Union, vol. 8, p. 19 (2012)Google Scholar
  26. 26.
    Zhang, Fupeng, Saha, Prasenjit: Probing the spinning of the massive black hole in the galactic center via pulsar timing: a full relativistic treatment. Astrophys. J. 849, 33 (2017)ADSCrossRefGoogle Scholar
  27. 27.
    Wang, Yan, Jenet, Frederick A., Creighton, Teviet, Price, Richard H.: Strong field effects on pulsar arrival times: circular orbits and equatorial beams. Astrophys. J. 697, 237–246 (2009)ADSCrossRefGoogle Scholar
  28. 28.
    Wang, Y., Creighton, T., Price, R.H., Jenet, F.A.: Strong field effects on pulsar arrival times: general orientations. Astrophys. J. 705, 1252–1259 (2009)ADSCrossRefGoogle Scholar
  29. 29.
    Edwards, R.T., Hobbs, G.B., Manchester, R.N.: TEMPO2, a new pulsar timing package—II. The timing model and precision estimates. MNRAS 372, 1549–1574 (2006)ADSCrossRefGoogle Scholar
  30. 30.
    Damour, T., Taylor, J.H.: Strong-field tests of relativistic gravity and binary pulsars. Phys. Rev. D 45, 1840–1868 (1992)ADSCrossRefGoogle Scholar
  31. 31.
    Damour, T., Deruelle, N.: General relativistic celestial mechanics of binary systems. II. The post-Newtonian timing formula. Ann. Inst. Henri Poincaré Phys. Théor. 44(3), 263–292 (1986)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Shapiro, I.I.: Fourth test of general relativity. Phys. Rev. Lett. 13, 789–791 (1964)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Blandford, R., Teukolsky, S.A.: Arrival-time analysis for a pulsar in a binary system. Astrophys. J. 205, 580 (1976)ADSCrossRefGoogle Scholar
  34. 34.
    Brumberg, V. A.: Essential relativistic celestial mechanics. Taylor & Francis, ISBN: 9780750300629Google Scholar
  35. 35.
    Zschocke, S., Klioner, S. A.: Analytical solution for light propagation in Schwarzschild field having an accuracy of 1 micro-arcsecond (2009). ArXiv e-prints Google Scholar
  36. 36.
    Schneider, J.: Gravitational beam deviation effects in the timing formula of binary pulsars. Astron. Astrophys. 232, 62 (1990)ADSGoogle Scholar
  37. 37.
    Doroshenko, O.V., Kopeikin, S.M.: Relativistic effect of gravitational deflection of light in binary pulsars. MNRAS 274, 1029–1038 (1995)ADSGoogle Scholar
  38. 38.
    Wex, N., Kopeikin, S.M.: Frame dragging and other precessional effects in black hole pulsar binaries. Astrophys. J. 514, 388–401 (1999)ADSCrossRefGoogle Scholar
  39. 39.
    Damour, T., Schäfer, G.: Higher-order relativistic periastron advances and binary pulsars. Il Nuovo Cimento B 101, 127 (1988)ADSCrossRefGoogle Scholar
  40. 40.
    Wex, N.: The second post-Newtonian motion of compact binary-star systems with spin. Class. Quantum Gravity 12, 983 (1995)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Königsdörffer, C., Gopakumar, A.: Post-Newtonian accurate parametric solution to the dynamics of spinning compact binaries in eccentric orbits: the leading order spin-orbit interaction. Phys. Rev. D 71, 024039 (2005)ADSCrossRefGoogle Scholar
  42. 42.
    Futamase, T., Itoh, Y.: The post-Newtonian approximation for relativistic compact binaries. Living Rev. Relativ. 10(1), 2 (2007)ADSCrossRefGoogle Scholar
  43. 43.
    Hartung, J., Steinhoff, J., Schäfer, G.: Next-to-next-to-leading order post-Newtonian linear-in-spin binary Hamiltonians. Ann. Phys. 525, 359 (2013)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Damour, T., Jaranowski, P., Schäfer, G.: Nonlocal-in-time action for the fourth post-Newtonian conservative dynamics of two-body systems. Phys. Rev. D 89, 064058 (2014)ADSCrossRefGoogle Scholar
  45. 45.
    Levi, M., Steinhoff, J.: Complete conservative dynamics for inspiralling compact binaries with spins at fourth post-Newtonian order (2016). ArXiv e-prints Google Scholar
  46. 46.
    Bernard, L., Blanchet, L., Bohé, A., Faye, G., Marsat, S.: Energy and periastron advance of compact binaries on circular orbits at the fourth post-Newtonian order. Phys. Rev. D 95(4), 044026 (2017)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    Semerák, O.: Approximating light rays in the Schwarzschild field. Astrophys. J. 800, 77 (2015)ADSCrossRefGoogle Scholar
  48. 48.
    Beloborodov, A.M.: Gravitational bending of light near compact objects. Astrophys. J. Lett. 566, L85–L88 (2002)ADSCrossRefGoogle Scholar
  49. 49.
    De Falco, V., Falanga, M., Stella, L.: Approximate analytical calculations of photon geodesics in the Schwarzschild metric. A&A 595, A38 (2016)ADSCrossRefGoogle Scholar
  50. 50.
    Lai, Dong, Rafikov, Roman R.: Effects of gravitational lensing in the double pulsar system J0737–3039. Astrophys. J. 621, L41–L44 (2005)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center of Applied Space Technology and Microgravity (ZARM)University of BremenBremenGermany
  2. 2.Faculty of PhysicsUniversity of BielefeldBielefeldGermany
  3. 3.Institute for Gravitation and the CosmosThe Pennsylvania State University, University ParkPennsylvaniaUSA
  4. 4.Department of PhysicsPennsylvania State UniversityUniversity ParkUSA
  5. 5.Department of PhysicsIndian Institute of Technology RoorkeeRoorkeeIndia

Personalised recommendations